55,442 research outputs found
Constructive topology of bishop spaces
The theory of Bishop spaces (TBS) is so far the least developed approach to constructive topology with points. Bishop introduced function spaces, here called Bishop spaces, in 1967, without really exploring them, and in 2012 Bridges revived the subject. In this Thesis we develop TBS.
Instead of having a common space-structure on a set X and R, where R denotes the set of
constructive reals, that determines a posteriori which functions of type X -> R are continuous with respect to it, within TBS we start from a given class of "continuous" functions of type X -> R that determines a posteriori a space-structure on X. A Bishop space is a pair (X, F),
where X is an inhabited set and F, a Bishop topology, or simply a topology, is a subset of all functions of type X -> R that includes the constant maps and it is closed under addition, uniform limits and composition with the Bishop continuous functions of type R -> R.
The main motivation behind the introduction of Bishop spaces is that function-based concepts are more suitable to constructive study than set-based ones. Although a Bishop topology of functions F on X is a set of functions,
the set-theoretic character of TBS is not that central as it seems. The reason for this is Bishop's inductive concept of the least topology generated by a given subbase. The definitional clauses of a Bishop space, seen as inductive rules, induce the corresponding induction principle. Hence, starting with a
constructively acceptable subbase the generated topology is a constructively graspable set of functions exactly because of the corresponding principle. The function-theoretic character of TBS is also evident in the characterization of morphisms between Bishop spaces.
The development of constructive point-function topology in this Thesis takes two directions. The first is a purely topological one. We introduce and study, among other notions, the quotient,
the pointwise exponential, the dual, the Hausdorff, the completely regular, the 2-compact,
the pair-compact and the 2-connected Bishop spaces. We prove, among other results, a Stone-Cech theorem, the Embedding lemma, a generalized version of the Tychonoff embedding theorem for completely regular Bishop spaces, the Gelfand-Kolmogoroff theorem for fixed and completely regular Bishop spaces, a Stone-Weierstrass theorem for pseudo-compact Bishop spaces and a Stone-Weierstrass theorem for pair-compact Bishop spaces. Of special importance is the notion of 2-compactness, a constructive function-theoretic notion of compactness for which we show that it generalizes the notion of a compact metric space. In the last chapter we initiate the basic homotopy theory of Bishop spaces.
The other direction in the development of TBS is related to the analogy between a Bishop topology F, which is a ring and a lattice, and the ring of real-valued continuous functions C(X) on a topological space X. This analogy permits a direct "communication" between TBS and the theory of rings of continuous functions, although due to the classical set-theoretic character of C(X) this does not mean a direct translation of the latter to the former. We study the zero sets of a Bishop space and we prove the Urysohn lemma for them. We also develop the basic theory of embeddings of Bishop spaces in parallel to the basic classical theory of embeddings of rings of continuous functions and we show constructively
the Urysohn extension theorem for Bishop spaces.
The constructive development of topology in this Thesis is within Bishop's informal system of constructive mathematics BISH, inductive definitions with rules of countably many premises included
Localic completion of uniform spaces
We extend the notion of localic completion of generalised metric spaces by
Steven Vickers to the setting of generalised uniform spaces. A generalised
uniform space (gus) is a set X equipped with a family of generalised metrics on
X, where a generalised metric on X is a map from the product of X to the upper
reals satisfying zero self-distance law and triangle inequality.
For a symmetric generalised uniform space, the localic completion lifts its
generalised uniform structure to a point-free generalised uniform structure.
This point-free structure induces a complete generalised uniform structure on
the set of formal points of the localic completion that gives the standard
completion of the original gus with Cauchy filters.
We extend the localic completion to a full and faithful functor from the
category of locally compact uniform spaces into that of overt locally compact
completely regular formal topologies. Moreover, we give an elementary
characterisation of the cover of the localic completion of a locally compact
uniform space that simplifies the existing characterisation for metric spaces.
These results generalise the corresponding results for metric spaces by Erik
Palmgren.
Furthermore, we show that the localic completion of a symmetric gus is
equivalent to the point-free completion of the uniform formal topology
associated with the gus.
We work in Aczel's constructive set theory CZF with the Regular Extension
Axiom. Some of our results also require Countable Choice.Comment: 39 page
Integrals and Valuations
We construct a homeomorphism between the compact regular locale of integrals
on a Riesz space and the locale of (valuations) on its spectrum. In fact, we
construct two geometric theories and show that they are biinterpretable. The
constructions are elementary and tightly connected to the Riesz space
structure.Comment: Submitted for publication 15/05/0
A Lindenstrauss theorem for some classes of multilinear mappings
Under some natural hypotheses, we show that if a multilinear mapping belongs
to some Banach multlinear ideal, then it can be approximated by multilinear
mappings belonging to the same ideal whose Arens extensions simultaneously
attain their norms. We also consider the class of symmetric multilinear
mappings.Comment: 11 page
Bounded holomorphic functions attaining their norms in the bidual
Under certain hypotheses on the Banach space , we prove that the set of
analytic functions in (the algebra of all holomorphic and
uniformly continuous functions in the ball of ) whose Aron-Berner extensions
attain their norms, is dense in . The result holds also for
functions with values in a dual space or in a Banach space with the so-called
property . For this, we establish first a Lindenstrauss type theorem
for continuous polynomials. We also present some counterexamples for the
Bishop-Phelps theorem in the analytic and polynomial cases where our results
apply.Comment: Accepted in Publ. Res. Inst. Math. Sc
Noncommutative Choquet theory
We introduce a new and extensive theory of noncommutative convexity along
with a corresponding theory of noncommutative functions. We establish
noncommutative analogues of the fundamental results from classical convexity
theory, and apply these ideas to develop a noncommutative Choquet theory that
generalizes much of classical Choquet theory.
The central objects of interest in noncommutative convexity are
noncommutative convex sets. The category of compact noncommutative sets is dual
to the category of operator systems, and there is a robust notion of extreme
point for a noncommutative convex set that is dual to Arveson's notion of
boundary representation for an operator system.
We identify the C*-algebra of continuous noncommutative functions on a
compact noncommutative convex set as the maximal C*-algebra of the operator
system of continuous noncommutative affine functions on the set. In the
noncommutative setting, unital completely positive maps on this C*-algebra play
the role of representing measures in the classical setting.
The continuous convex noncommutative functions determine an order on the set
of unital completely positive maps that is analogous to the classical Choquet
order on probability measures. We characterize this order in terms of the
extensions and dilations of the maps, providing a powerful new perspective on
the structure of completely positive maps on operator systems.
Finally, we establish a noncommutative generalization of the
Choquet-Bishop-de Leeuw theorem asserting that every point in a compact
noncommutative convex set has a representing map that is supported on the
extreme boundary. In the separable case, we obtain a corresponding integral
representation theorem.Comment: 81 pages; minor change
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